Orthogonal and Orthonormal Sets

 Definition:

 Let \(\pmb{V}\) be a inner product space over a field \(\pmb{F}\) and let \(\pmb{x,y}\) be two vectors in \(\pmb{V}\). The vectors \(\pmb{x}\) and \( \pmb{y}\) are said to be \(\fcolorbox{blue}{white}{\pmb{orthogonal}}\) if \( \fcolorbox{blue}{white}{\(\pmb{\lang x,y \rang =0}\)}\).

 Example:

  Let \(\pmb{x,y\in \mathcal{R}^{2}}\) such that \(\pmb{x=(1,2)}\) and \(\pmb{y=(2,-1)}\) then \(\pmb{\lang x,y \rang=1\centerdot 2 + 2\centerdot (-1)=0}\). Therefore the vectors are \(\pmb{x,y}\) are orthogonal.

 Definition:

 Let \(\pmb{V}\) be a inner product space over a field \(\pmb{F}\) and let \(\pmb{S\subset V}\). Then \(\pmb{S}\) is said to be an \( \fcolorbox{blue}{white}{\pmb{orthogonal~set}}\) if \(\pmb{\lang x,y \rang=0}\) \(\pmb{\forall~ x,y\in R}\) where \(\pmb{x\ne y}\) .

 Example:

  Let \(\pmb{x,y\in \mathcal{R}^{2}}\) such that \(\pmb{x=(1,2)}\) and \(\pmb{y=(2,-1)}\) and let \(\pmb{S=\{x,y\}}\) then \(\pmb{S}\) is orthogonal.

 Definition:

 Let \(\pmb{V}\) be a inner product space over a field \(\pmb{F}\) and let \(\pmb{x\in V}\). Then \(\pmb{x}\) is said to be an \( \fcolorbox{blue}{white}{\pmb{unit~vector}}\) if \(\fcolorbox{blue}{white}{\pmb{||x||=1}}\) i.e., \(\pmb{\lang x,x \rang =1}\).

 Example:

  Let \(\pmb{x,y\in \mathcal{R}^{2}}\) such that \(\pmb{x=(\frac{1}{\sqrt{5}},\frac{2}{\sqrt{5}})}\) and \(\pmb{y=(\frac{2}{\sqrt{5}},-\frac{1}{\sqrt{5}})}\) then \(\pmb{||x||=1}\) and \(\pmb{||y||=1}\). Therefore \(\pmb{x}\) and \(\pmb{y}\) are unit vectors.

 Definition:

 Let \(\pmb{V}\) be a inner product space over a field \(\pmb{F}\) and let \(\pmb{S\subset V}\). Then \(\pmb{S}\) is said to be an \( \fcolorbox{blue}{white}{\pmb{orthonormal~set}}\) if
 (i)   \(\pmb{||x||=1}\) \(\pmb{\forall~ x\in R}\)
 (ii)   \(\pmb{\lang x,y \rang=0}\) \(\pmb{\forall~ x,y\in R}\) where \(\pmb{x\ne y}\).
i.e.,
 (i)   Each vectors of \(\pmb{S}\) is an unit vector.
 (ii)  \(\pmb{S}\) is an orthogonal set.

 Example:

  Let \(\pmb{x,y\in \mathcal{R}^{2}}\) such that \(\pmb{x=(\frac{1}{\sqrt{5}},\frac{2}{\sqrt{5}})}\) and \(\pmb{y=(\frac{2}{\sqrt{5}},-\frac{1}{\sqrt{5}})}\) and let \(\pmb{S=\{x,y\}}\) then \(\pmb{S}\) is an orthonormal set.